Can a Facial Implant Move Around in Your Face?

If a cheek implant is placed in the right plane (i.e. the subperiosteal plane), it will not move. It is usually fixed in place by the surrounding tissues within 48-72 hours of insertion. After that, silicone implants are encapsulated and will not move. 

A facial implant that is not secured can shift within the pocket in which it was placed. In some cases, your surgeon may affix the implant with a screw in to the bone or a suture to covering of the bone to prevent shifting.
Than answer to your second question depends on several variables (whether to get fat or implants). There are pluses and minues either way.
Facial implants are permanent but they can get infected or shift (as mentioned above). Fat injections are soft and can be more natural looking, especially if you have very thin skin and tissues, but the fat may or may not be permanent.
Be sure to discuss with your doctor the benefits and drawbacks of each option, based on your particularly anatomy and desire outcome.

Facial implants are placed in precise pockets deep to the periosteum. It is rare for them to move ,without trauma, after one week. Some surgeons choose to fix them with a screw, but I don't feel this is necessary. Fat augmentation is an excellent alternative approach.